## Orders in self-injective cogenerator rings

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- by Robert C. Shock
- Proc. Amer. Math. Soc.
**35**(1972), 393-398 - DOI: https://doi.org/10.1090/S0002-9939-1972-0302683-6
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## Abstract:

This note states necessary and sufficient conditions for a ring to be a right order in certain self-injective rings. A ring*R*is said to have the dense extension property if each

*R*-homomorphism from a right ideal of

*R*into

*R*can be lifted to some dense right ideal of

*R*. A right ideal

*K*is rationally closed if for each $x \in R - K$ the set ${x^{ - 1}}K = \{ y \in R:xy \in K\}$ is not a dense right ideal of

*R*. We state a major result. Let $\dim R$ denote the Goldie dimension of a ring

*R*and $Z(R)$ the right singular ideal of

*R*. Then

*R*is a right order in a self-injective cogenerator ring if and only if

*R*has the dense extension property, $Z(R)$ is rationally closed and the factor ring $R/Z(R)$ is semiprime with $\dim R/Z(R) = \dim R < \infty$.

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## Bibliographic Information

- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**35**(1972), 393-398 - MSC: Primary 16A18
- DOI: https://doi.org/10.1090/S0002-9939-1972-0302683-6
- MathSciNet review: 0302683